SCF methods

With radicals there is no convenient method like the Hartree-Fock-Roothaan procedure commonly used for closed-shell systems. In contrast, the open-shell theory is typical of a number of methods suggested which differ in accuracy from the viewpoint of true SCF theory, in range of applicability, complexity, and computing feasibility. A critical survey of open-shell SCF methods reported by Berthier D covers the literature up to 1962. We shall not duplicate that review here we propose rather to note some features of open-shell methods relevant to their computation feasibility and to mention procedures published after 1962. The unrestricted treatments that assume different space orbitals for different spins will be disregarded here because the restricted wave functions 

When discussing open-shell methods, it is convenient to use the formalism of the well-known Roothaan procedure 3). Hence, consider an open-shell configuration for which the total energy can be expressed as 

The density-matrix method of McWeeny was found to converge in all cases, but too slowly for practical purposes. The energy-weighted steepest descent (EWSD) method developed by Hillier and Saunders, is related to the McWeeny method it is claimed that the modifications adopted bring about improved convergence. 

The last group in our classification comprises two approximate SCF procedures which give wave functions that are not correct to first order. The first of them, Nesbet s method of symmetry and equivalence restrictions uses the Hamiltonian of the unrestricted method for the a-spin electrons, the number of a-spin electrons being greater than that of p-spin electrons. The p-spin electrons are forced to occupy MOs given for a-spin electrons by 

The second method of this group, the method of Longuet-Higgins and Pople 20), with its extension for accommodating states with degenerate open shells is described in detail in the next section. 

---

Roos B O, Taylor P R and Siegbahn P E M 1980 A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach Chem. Phys. 48 157-73... 

Bacskay G B 1981 A quadratically convergent Hartree-Fock (QC-SCF) method. Applications to the closed-shell case Chem. Phys. 61 385... 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

Sin glc-poiri t, georn ctry oplim i/ation, molecular dynam ies an d vibration calctilalioti s are all available with either ah initio or semi-empirical SCf methods.. After obtain in g a wavefiincLion via any of... 

In the Huckel theory of simple hydrocarbons, one assumes that the election density on a carbon atom and the order of bonds connected to it (which is an election density between atoms) are uninfluenced by election densities and bond orders elsewhere in the molecule. In PPP-SCF theory, exchange and electrostatic repulsion among electrons are specifically built into the method by including exchange and electrostatic terms in the elements of the F matrix. A simple example is the 1,3 element of the matrix for the allyl anion, which is zero in the Huckel method but is 1.44 eV due to election repulsion between the 1 and 3 carbon atoms in one implementation of the PPP-SCF method. 

Using more flexible trial funetions (polynomials in r, Y2, and ri2 perhaps) for /(ri), one ean ealeulate very aeeurate energies for He by the SCF method. This gives us eonfidenee that we ean make a valid generalization to larger systems. 

Having the Slater atomic orbitals, the linear combination approximation to molecular orbitals, and the SCF method as applied to the Fock matrix, we are in a position to calculate properties of atoms and molecules ab initio, at the Hartree-Fock level of accuracy. Before doing that, however, we shall continue in the spirit of semiempirical calculations by postponing the ab initio method to Chapter 10 and invoking a rather sophisticated set of approximations and empirical substitutions... 

Once the format of the Fock matrix is known, the semiempirical molecular problem (and it is a considerable one) is finding a way to make valid approximations to the elements in the Fock matrix so as to avoid the many integrations necessary in ab initio evaluation of equations like Fij = J 4>,F4> dx. After this has been done, the matrix equation (9-62) is solved by self-consistent methods not unlike the PPP-SCF methods we have already used. Results from a semiempirical... 

Introductory descriptions of Hartree-Fock calculations [often using Rootaan s self-consistent field (SCF) method] focus on singlet systems for which all electron spins are paired. By assuming that the calculation is restricted to two electrons per occupied orbital, the computation can be done more efficiently. This is often referred to as a spin-restricted Hartree-Fock calculation or RHF. 

C in D2O. Extensive tables are found in Lehn s review <70MI50100). Calculations of inversion barriers have met with mixed success. The MNDO SCF method gives results which compare well with experimental values, including the high barriers of Af-halo- and N-amino-aziridines, and the low ones for Af-trimethysilyl- and Af-phosphino-aziridines <80JCS(P2)1512). 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

One of the most efficient ways to treat this problem is to combine the ab initio MO method and the RISM theory, and this has been achieved by a slight modification of the original RISM-SCF method. Effective atomic charges in liquid water are determined such that the electronic structure and the liquid properties become self-consistent, and along the route of convergence the polarization effect can be naturally incorporated. 

Ab initio calculations are iterative procedures based on self-consistent field (SCF) methods. Normally, calculations are approached by the Hartree-Fock closed-shell approximation, which treats a single electron at a time interacting with an aggregate of all the other electrons. Self-consistency is achieved by a procedure in which a set of orbitals is assumed, and the electron-electron repulsion is calculated this energy is then used to calculate a new set of orbitals, which in turn are used to calculate a new repulsive energy. The process is continued until convergence occurs and self-consistency is achieved." ... 

In general conclusion, the HMO and SCF methods both appear able to make reasonably accurate predictions about the stabilization in conjugated moleeules. The stabilization is general for benzenoid compounds but quite restricted in nonbenzenoid systems. Because the HMO method of estimating stabiUty is based on the ideas of HMO theory, its general success vindicates the ability of this very simplified MO approach to provide insight into the structural nature of the aimulenes and other conjugated polyenes. More sophisticated MO methods, of course, are now accessible and should be applied for more detailed analysis of the structures of these molecules. 

The value is in hartrees. The number of cycles it took the SCF calculation to converge is also given on this line (refer to Appendix A for a discussion of the iterative nature of the SCF method). When we discuss energies in this work, we will generally use hartrees (atomic units) when we discuss energy differences, kcal-moT will often be a more convenient unit (especially when comparing calculation predictions to experimental results). 

Selufien The basic strategy behind the direct SCF method is recomputing certain intermediate quantities within the calculation—specifically the two-electron integrals—as needed, rather than storing them on disk. This has the advantage of making it possible to study systems which would require more disk space than is available on the system. 

A variety of theoretical methods have been developed which include some effects of electron correlation. Traditionally, such methods are referred to as post-SCF methods because they add correlation corrections to the basic Hartree-Fock model. As of this writing, there are many correlation methods available in Gaussian, including the following ... 

Methods based on Density Functional Theory also include some electron correlation effects (we ll consider them a bit later in this chapter). Of the traditional post-SCF methods, we ll be primarily using MP2, MP4, QCISD and QCISDfO in this work. 

SCF) method. At convergence, the energy is at a minimum, and the orbitals generate a field which produces the same orbitals, accounting for the method s name. The solution produces a set of orbitals, both occupied and virtual (unoccupied,... 

The general strategy used by the SCF method (after initial setup steps) is as follows ... 

In the Direct SCF method, we do. not store the two-electron integrals over the basis functions, we recalculate them on demand every cycle of the HF procedure At first sight, this may seem wasteful, but Conventional methods rely on disk input/output transfer rates whilst Direct methods rely on processor power. There is obviously a balance between processor speed and disk I/O. Just for the record my calculation on aspirin (73 basis functions) took 363 s using the Direct method and 567 s using the Conventional method. 

Sir William Hartree developed ingenious ways of solving the radial equation, and they are documented in Douglas R. Hartree s book (1957). By the time this book was published, the SCF method had been well developed, and its connection with the variation principle was finally understood. It is interesting to note that Chapter 2 of Douglas R. Hartree s book deals with the variation principle. 

---